1. Application of Finite Geometry LDPC code on the Internet Data Transport Wu Yuchun Oct 2006 Huawei Hisi Company Ltd

2. 2 Contents Abstract. Simple Encoding Algorithm Simple Decoding Algorithm Other Advantages Applied in the Erasure Channel End

3. 3 Abstract In this file we propose a Finite Geometry LDPC(FG_LDPC) code to be applied in the Internet data transport because of several of its advantages which we think are very important for the huge amount of data transport.

4. 4 Simple Encoding Algorithm The Encoding of FG_LDPC code is very simple. It’s just like the Encoding of a cyclic code. Suppose we denote Message Polynomial Generator Polynomial Then the output of the Encoder is just

5. 5 Simple Decoding Algorithm. （ 1/4 ） Some of FG_LDPC Codes are One Step Majority Logical Decodable Codes, so the decoding algorithm is very simple if compared to other LDPC decoding algorithms such as Sum Product Algorithm, Bit Flip Algorithm…. The decoding procedure can be summarized as follows: 1)

6. 6 Simple Decoding Algorithm （ 2/4 ） Where is the received vector (one row of the Received Matrix as explained in Slide11-13) and H is the Parity Check Matrix of FG_LDPC Code. We call as Syndrome Vector. 2) Calculate the Error Vector For every element, we define a position set A(i), which consists the row indexs of H that the i th element of that row is “1”. then if more than half of s(A(i)) are 1, we decided that, i.e, the i th received element is in error, otherwise, the i th received element is correct and

7. 7 Simple Decoding Algorithm （ 3/4 ） For Example, for the Euclidean Finite Geometry The corresponding Parity Check Matrix is

8. 8 Simple Decoding Algorithm （ 4/4 ） For and, from H we can see that the So if more than half of the Syndrome element of is “1”,, otherwise, If more than half of the Syndrome element of is “1”,, otherwise,

9. 9 Other Advantages （ 1/2 ） There is an ample selection of block lengths and code rates of FG_LDPC Code; Along with the increase of Code Length, the Encoding and Decoding Complexity only increase linearly

10. 10 Other Advantages （ 2/2 ） For some specific Finite Geometry, the Parity Check Matrix H can be put in a cyclic form, any row of the H is only a cyclic shifted version of the first row. So in the real application, we only need to save one row of H rather than the whole Matrix H. In real application, we can calculate the g(x) and H off line and save it, then just use it when the transport begin. This will further reduce the CPU load on the transport procedure ；

11. 11 Applied in the Erasure Channel （ 1/7 ） The application of this FG_LDPC in the Erasure Channel is illustrated in the following figure:

12. 12 Applied in the Erasure Channel （ 2/7 ） In the sender side :  Step1: Write all data of a block into a Source Matrix column by column. Each Source Symbol correspond to a column of the Source Matrix, Each Source Vector correspond to a row of the Source Matrix;  Step2: Encoding was done on the Source Matrix row by row. Each Source Vector(k bits) was encoded into a Encoding Vector(n bits). after encoding, we obtain the Encoding Matrix with dimension of ((E*8) * n);  Step3: Each column of the Encoding Matrix was considered as an encoding symbol, transmit the Encoding Matrix column by column.

13. 13 Applied in the Erasure Channel （ 3/7 ） In the receiver side :  Step1: Arrange all received symbols of a block into a Receive Matrix. Each received symbol correspond to a column of the Receive Matrix.  Step2: Decode the Receive Matrix row by row, and obtain the Decoded Matrix with dimension of ((E*8) * k) ;  Step3: Output the Decoded Matrix column by column.

14. 14 Applied in the Erasure Channel （ 4/7 ） To be simple, for the transmission, the data is transmitted symbol by symbol (or column by column); for encoding and decoding, the data is in a vector by vector (or row by row) order. For each encoding and decoding operation, it’s just the operation as described in Slide 4 and Slide 5-8. We can see that each Erasured symbol only contribute one Erasured bit for the decoding unit. because the data fed into the decoder is a row vector. So it’s just like the interleaver: disturb the burst erasure (Erasured Symbol) into a lot of Bit Erasures.

15. 15 Applied in the Erasure Channel （ 5/7 ） Another Encoding method to apply FG_LDPC in Erasure Channel are as follow figure:

16. 16 Applied in the Erasure Channel （ 6/7 ） The Encoding Procedure is :  Step1: All the registers are set to 0;  Step2: Key K 0 connect to point A 00 and K 1 connect to point A 01, and the source symbols are input in the order of U 0,U 1,U 2, …U k-1,  Step3: After k clock cycles, all k Source Symbols was finished, we connect Key K 0 to point A 01 and K 1 to point A 11, after another n-k clock cycles, all n-k parity symbols was outputed.  Then the encoding was finished, the Output symbols are U 0, U 1, U 2, …U k-1, P 0, P 1, P 2, …P n-k-1. Each symbol is E Bytes wide.

17. 17 Applied in the Erasure Channel （ 7/7 ） Another Decoding Algorithm of FG_LDPC in Erasure Channel can be found in Chapter 6.4 of 《 draft-ietf-rmt-bb-fec-ldpc- 03 》 :  For every Erasure Symbol, searching on all related check equations, if there is an equation in which only this Erasure Symbol is a variable, then we can recover this Erasure Symbol;  Replacing these Recovered Erasure Symbols in all check equations, and continue to Recover remained Erasure Symbols in the same way iteratively,  If in some step, For any one of the remained Erasure Symbols, we can’t find a check equation in which only this Erasure Symbol is a variable, we finished decoding and report that this block of symbols was failed to be recovered, otherwise, we can recover this block of symbols successfully.

18. 18 End Any comments and questions are welcomed and appreciated. We hope to work together with people who are interest in it so that we can improve and perfect this FEC scheme on the Internet data transport.

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